Precalculus Examples

$\ln (x + 9) = \ln (x) + \ln (9)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $\ln (x) + \ln (9)$ .

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Step 1.1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (x + 9) = \ln (x \cdot 9)$

Step 1.1.2

Move $9$ to the left of $x$ .

$\ln (x + 9) = \ln (9 x)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$x + 9 = 9 x$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $9 x$ from both sides of the equation.

$x + 9 - 9 x = 0$

Step 3.1.2

Subtract $9 x$ from $x$ .

$- 8 x + 9 = 0$

Step 3.2

Subtract $9$ from both sides of the equation.

$- 8 x = - 9$

Step 3.3

Divide each term in $- 8 x = - 9$ by $- 8$ and simplify.

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Step 3.3.1

Divide each term in $- 8 x = - 9$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 9}{- 8}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 8$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 8 x}{- 8} = \frac{- 9}{- 8}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 9}{- 8}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{9}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{9}{8}$

Decimal Form:

$x = 1.125$

Mixed Number Form:

$x = 1 \frac{1}{8}$